Kernelization

1. MashrurAlam Khan School of Computer Science University of Waterloo Canada Kernelization

2. Outline • Background Materials • A Closer Look at Some Parameterized Problems • p-MAX-3-SAT • p-FAST • p-d-Hitting Set • p-Dominating Set on Planar Graph • p-Max Leaf Tree • Conclusions and Future Work

3. Tractability • For tractable problems, the running time of the program is a polynomial function of the input size n, say nc • For intractable problems, it is an exponential function of cn • Usually, for constant c and large n, cn> nc

4. How to deal with Intractability • Find an improved algorithm • Unlikely, because most computer scientists believe that P ≠ NP • Or, associate a parameter with each intractable problem • Introduction to parameterized algorithm

5. The Basic Idea Brute Force Approach O (const|x|) Input instance, x Solution Parameter, k An unique mapping from (x, k) to (x’, k’) Runtime = map(x) * const|x’| Brute Force Approach O (const|x’|) Input instance (x’, k’) Solution

6. Kernelization • The transformation from (x, k) to (x’, k’) • Obviously, the size of |x’| must not be dependant on the size of |x| anymore • Formally, • |x’| ≤ |kernel_size(k)|, where g is a function • (x’, k’) Єinput_set <=> (x, k) Єinput_set • Also, |k’| ≤ |k|

7. Types of Kernelization • Determined from the function of the size of kernel, kernel_size • Linear • Polynomial • Exponential (to be covered in the advanced kernelization seminar)

8. Fixed Parameter Tractability • Runtime • O (map(x) * const|x’|) • ≤ O (map(x) * const|kernel_size(k)|) • Fixed Parameter Tractability • p-Vertex Cover • Non-FPT • p-Dominating Set, p-Clique • Beyond the scope of the presentation • We will stick to FPT, linear and polynomial kernels

9. Outline • Background Materials • A Closer Look at Some Parameterized Problems • p-MAX-3-SAT • p-FAST • p-d-Hitting Set • p-Dominating Set on Planar Graph • p-Max Leaf Tree • Conclusions and Future Work

10. p-MAX-3-SAT • Problem • (a1V a2V a3) ^ (a1V a2V a4) ^ … ^ (a6V a8V an) • n variables, m clauses, parameter = k • Input size, |x| = m ≤ 3n • Is there any assignment of variables which satisfies at least k of these clauses? 1 2 m

11. p-MAX-3-SAT • Observation • Consider any arbitrary assignment • a1= 1, a2= 1, a3= 0, …, an - 1 = 1, an= 0 • Does it satisfy m/2 clauses? • If yes, move to the next slide • If no, then its complement will • a1= 0, a2= 0, a3= 1, …, an - 1 = 0, an= 1

12. p-MAX-3-SAT • If k < m / 2 • Trivially yes • If k ≥ m / 2 • m ≤ 2k • Delete the variables that do not occur in any clause • n ≤ 6k, because each clause contain at most three new variables • We have obtained a kernel of size 6k • Linear kernel

13. Outline • Background Materials • A Closer Look at Some Parameterized Problems • p-MAX-3-SAT • p-FAST • p-d-Hitting Set • p-Dominating Set on Planar Graph • p-Max Leaf Tree • Conclusions and Future Work

14. p-FAST-Background • Tournament • A directed graph where each pair of vertices are connected by an arc • Feedback Arc Set • A set of arcs whose removal turns a tournament into an acyclic graph 1 2 3 4 5

15. p-FAST

16. Outline • Background Materials • A Closer Look at Some Parameterized Problems • p-MAX-3-SAT • p-FAST • p-d-Hitting Set • p-Dominating Set on Planar Graph • p-Max Leaf Tree • Conclusions and Future Work

17. p-d-Hitting Set • Universe, U • {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} • Family of sets with fixed cardinality, F • {{0, 1, 2}, {4, 0, 2}, {0, 6, 1}, …, {3, 8, 7}} • Cardinality of sets within F, d • 3 • Parameter, k • Is there a subset of U, U’? • U’ = {0, 1, …, 3} and |U’| ≤ k

18. p-d-Hitting Set • Sunflower • A collection of sets with common intersection • S = {S1, S2, …, Sk} • S1Ω S2 = S2Ω S3 = S1Ω S3 = … = SiΩSj = … = Sk-1ΩSk = C • Assume we have a sunflower S in F, |S| = k + 1 • Throw away the unnecessary parts and obtain new F’ • <U, F, k> = <U, F’, k> • Kernel size will be O(kd * d!) sets (Theorem)

19. Outline • Background Materials • A Closer Look at Some Parameterized Problems • p-MAX-3-SAT • p-FAST • p-d-Hitting Set • p-Dominating Set on Planar Graph • p-Max Leaf Tree • Conclusions and Future Work

20. p-Dominating Set on Planar Graph • Given a planar graph G (V, E) and a parameter k, is there a dominating set of size at most k in G? • Data reduction rule (Produces linear kernel) • For any vertex, v,

21. Outline • Background Materials • A Closer Look at Some Parameterized Problems • p-MAX-3-SAT • p-FAST • p-d-Hitting Set • p-Dominating Set on Planar Graph • p-Max Leaf Tree • Conclusions and Future Work

22. p-Max Leaf Spanning Tree • Given a graph G (V, E) and a parameter k, find a spanning tree T in that graph with at least k leaves (if exists). • Data reduction rule (Produces linear kernel)

23. Outline • Background Materials • A Closer Look at Some Parameterized Problems • p-MAX-3-SAT • p-FAST • p-d-Hitting Set • p-Dominating Set on Planar Graph • p-Max Leaf Tree • Conclusions and Future Work

24. Conclusions and Future Work • We have seen • Kernelization basics • A few examples on how it works • We will see (in the advanced kernelization seminar) • Exponential Kernels • Bounds on kernel size • Lower • Upper • Recent developments on the bounds

25. Questions? Thank You